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"Adding" 2 open sets
I'm trying to prove that If both S and T are open sets then S+T is open set as well.
[itex]S+T=\{s+t \| s \in S, t \in T\}[/itex]
S+T is open if every point [itex] x_0 \in S+T [/itex] is inner point.
Let [itex]x_0[/itex] be a point in S+T, so there is [itex]s_0[/itex] in S and [itex]t_0[/itex] in T so that [itex]x_0=s_0+t_0[/itex].
S is open so for every ||s-s_0|| < δ_1 s in S.
T is open so for every ||t-t_0|| < δ_2 t in T.
Let x be point in S([itex]x_0[/itex], _delta_), I will write x=s+t. [both s and t are some vectors in R^n]
s+t in S([itex]x_0[/itex], _delta_)={s+t | ||[itex]s+t-s_0-t_0[/itex]|| < _delta_} and here I stuck, if I could conclude from ||[itex]s+t-s_0-t_0[/itex]|| < _delta_ that ||[itex]s-s_0[/itex]|| < δ_1 and ||[itex]t-t_0[/itex]|| < δ_2 the proof will be over, however I just can't find the algebraic manipulation.
Will appreciate any help.
Thanks.
Homework Statement
I'm trying to prove that If both S and T are open sets then S+T is open set as well.
Homework Equations
[itex]S+T=\{s+t \| s \in S, t \in T\}[/itex]
The Attempt at a Solution
S+T is open if every point [itex] x_0 \in S+T [/itex] is inner point.
Let [itex]x_0[/itex] be a point in S+T, so there is [itex]s_0[/itex] in S and [itex]t_0[/itex] in T so that [itex]x_0=s_0+t_0[/itex].
S is open so for every ||s-s_0|| < δ_1 s in S.
T is open so for every ||t-t_0|| < δ_2 t in T.
Let x be point in S([itex]x_0[/itex], _delta_), I will write x=s+t. [both s and t are some vectors in R^n]
s+t in S([itex]x_0[/itex], _delta_)={s+t | ||[itex]s+t-s_0-t_0[/itex]|| < _delta_} and here I stuck, if I could conclude from ||[itex]s+t-s_0-t_0[/itex]|| < _delta_ that ||[itex]s-s_0[/itex]|| < δ_1 and ||[itex]t-t_0[/itex]|| < δ_2 the proof will be over, however I just can't find the algebraic manipulation.
Will appreciate any help.
Thanks.
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